A 6'5" center from Winfield, Kansas, Brannum attended the University of Kentucky and Michigan State before playing professional basketball.

In Brannum's final season with Sheboygan, when the Redskins finished with the NPBL's best record, he was selected first-team center after having the league's high scoring average (19.0 points per game).

Brannum's 45-point barrage against the Kansas City Hi-Spots on December 28, 1950, was a franchise record, topping the 44 points Bobby Cook scored against the NBA's Denver Nuggets the previous January.

Brannum often served as an unofficial "bodyguard" for smaller players on the team, especially point guard Bob Cousy.

Cousy later remarked in an interview, "It was a great luxury to have Bob on the team, and to have him playing the role of protector.